Spread complexity and topological transitions in the Kitaev chain

• Published in: The journal of high energy physics Vol. 2023; no. 1; pp. 120 - 30
• Main Authors: Caputa, Pawel, Gupta, Nitin, Haque, S. Shajidul, Liu, Sinong, Murugan, Jeff, Van Zyl, Hendrik J. R.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 23.01.2023; Springer Nature B.V; SpringerOpen
• Subjects: AdS-CFT Correspondence; Chains; Classical and Quantum Gravitation; Complexity; Critical point; Elementary Particles; Fermions; High energy physics; Holography and Condensed Matter Physics (AdS/CMT); P waves; Phase Transitions; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Topological States of Matter; Topology; AdS-CFT Correspondence; Topological States of Matter; Holography and Condensed Matter Physics (AdS/CMT); Phase Transitions
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP01(2023)120

A bstract A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor — the Kitaev chain — as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.